The long-term aim of this project is the improvement of tomographic imaging in diagnostic radiology and nuclear medicine, especially positron emission tomography (PET). The project involves computer methods for generating images from the raw detector measurements of the PET scanner (i.e., algorithms for image reconstruction from projection data). The Fourier transform is an important mathematical tool in the theory of non- iterative reconstruction methods, but its potential for improving practical computation has not been fully exploited for these methods, and has not been investigated at all for iterative reconstruction methods. The project aims to develop and evaluate the usefulness of Fourier-based approaches for non-iterative and iterative reconstruction in 3D PET. The proposed approaches aim to decrease the number of approximations that are made during the processing of the emission data and transmission data measured by the detector system of the scanner, while requiring less scanning and processing time than the simplified approaches currently used in clinical PET. For the emission data, the specific aim is to test and evaluate a carefully studied and optimized implementation of the direct Fourier method for the geometry of fully-3D PET, and to develop, implement, test, and evaluate Fourier-based 3D iterative techniques utilizing information on attenuation, scatter and randoms within the reconstruction model. For the transmission data, the specific aim is to develop and evaluate fully-3D iterative reconstruction from this data, followed by fast Fourier-based reprojection of the attenuation map, to be used either for correction of the fully-3D emission data, or for modeling of attenuation effects within the fully-3D iterative reconstruction from the emission data. Cancer imaging with PET, especially whole-body scanning, is limited by the low numbers of counts obtainable in the emission and transmission data sets and by the large fractions of scatter and random events. Improving the quantitative accuracy, signal- to-noise performance and lesion detectability of clinically practical data acquisition and processing in PET would lead to improved detection, diagnosis, and treatment planning of cancer.